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Data processing: the Wigner-Ville distribution
Values of fundamental frequency of the buildings were computed at the beginning and during each seismic event. Using time-frequency distributions (Neild et al., 2003), we monitor instantaneous small frequency variations that occur over time in the presence of external perturbations. We used a Cohen’s class distribution (Cohen, 1989), also called energy distribution. The energy of a signal x(t) can be deduced from the squared modulus of either the signal or its Fourier transform, i.e.
We can interpret |x(t)|2 and |X(ω)|2 as energy densities, respectively in time t and in frequency ω. The time-frequency energy distribution is covariant by translation in time and in frequency (Cohen, 1989). This means that if we integrate the time-frequency energy density along one variable, we obtain the energy density corresponding to the other variable. Applied to buildings, where energy is carried by resonance modes, Cohen’s class tracks the energy variation at the considered modal frequency in time. In this study, we considered the Wigner-Ville (WV) distribution designed for the analysis of non-stationary signals, and defined thus:
We therefore considered a windowed version of the Wigner-Ville distribution, called the pseudo Wigner-Ville distribution, noted PWV, given by the following equation:
where h( 98 2 2 is a classical windowing function. This windowing is equivalent to a frequency smoothing ) of the WV function that attenuates oscillation of the interferences compared with WV. However, this formulation shows the classical time-frequency distribution trade-off between time and frequency resolutions: by selecting a short time window h, the smoothing function will be narrow in time and wide in frequency, and vice versa. PWV is then only controlled by the short-time window h(t). We can therefore add a degree of freedom to the distribution, by considering a separable smoothing function g(t), to allow progressive and independent control in both time and frequency of the smoothing applied to the PWV. This new distribution is called the smoothed-pseudo Wigner-Ville distribution (sPWV), as follows:
The previous time-frequency trade-off is now replaced by a compromise between the joint time-frequency resolution and the amplitude of the interferences. However, Eq. 1.4 shows that a time-frequency domain in the vicinity of each (t,ω) point is delimited, within which the weighted average of the time-frequency values is computed. The energy band of the distribution remains broad, making it difficult to distinguish slight variations of the frequency carrying the maximum energy. A reassignment method has been proposed (Kodera et al., 1976; Auger and Flandrin, 1995) to solve the trade-off between the reduction of misleading interferences and a sharp concentration of the signal component. This method states that there are no physical reasons why the WV values should be symmetrically distributed around each point (t,ω), which is considered as the geometrical center of the domain. Consequently, their average must be assigned at the center of gravity, which is much more representative of the local energy distribution of the signal than the geometric center of this domain. In practice, the reassignment method (δ operator) shifts( ,W Weach) value of the time-frequency distribution from any point (t,ω) to the center of gravity of the signal energy distribution around (t,ω). The mathematical implementation of the reassignment proposed by Kodera et al. (1976) is developed in Auger and Flandrin (1995) for several classical time-frequency methods. Equation 1.4 is then modified to the reassigned rsPWV distribution, whose value at any point (t’, ω’) is the sum of all the distribution values reassigned to this point, i.e.:
Several other time-frequency distribution methods are available in the literature but Michel and Guéguen (2010) compared several time-frequency distributions of accelerograms recorded in buildings, with and without reassignment. They concluded on the efficiency of the rsPWV distribution for our application, i.e., for tracking the fast dynamic variation of the energy position in resonance frequency during earthquakes, reducing both interferences and the aforementioned trade-off. In this study, rsPWV was applied automatically to the whole dataset, using a 4th order of decimation in frequency and 2,048 frequency points (N). Time h and frequency g smoothing windows were taken to be default Hamming windows, with N/10 and N/4 points, respectively. Furthermore, instantaneous maximum energy value variations are tracked using a 3rd order Savitzky-Golay (polynomial smoothing window) filter with a size equal to 15% of the triggered window (i.e. time window of the duration of the record). Figure 1.5 shows an example of the rsPWV distribution and the curve corresponding to the Savitzky-Golay filter. The characteristic values of the resonance frequency observed during each earthquake are automatically picked (and manually checked) from this curve for efficient application to the whole dataset.
Figure 1.5 a) Time-frequency distribution during an earthquake, computed using the Wigner-Ville distribution. b) Reassigned smoothed-pseudo Wigner-Ville distribution (i.e. rsPWVx) applied to the plot in a). The solid black line corresponds to the Savitzky-Golay function. Pre-seismic fi and co-seismic fmin fundamental frequencies are indicated. The color scale represents the energy intensity. c) Acceleration time history at the top of the building, with idx_fi as the first arrival time.
The pre-seismic value fi associated with the elastic fundamental frequency is computed by the Fourier transform of the zero-padded (16,384 samples) pre-event noise window. The first arrival time (i.e. idx_fi) of the earthquake is obtained by applying the STA/LTA algorithm (Short Term Average to Long Term Average), with STA/LTA = 2.
fmin corresponds to the co-seismic minimum value of the fundamental frequency computed from the time-frequency distribution of each record. Indeed, co-seismic variations of frequency are reported as related to structural health and nonlinear processes, generally associated with the opening/closing of preexisting cracks (i.e. Clinton et al., 2006; Todorovska and Trifunac, 2007; Astorga et al, 2018, 2019). In the flat-file, fmin corresponds to the average value of ± 10 samples around the minimum value observed in the smoothing function
Structure of the NDE1.0 flat-file
The flat-file is divided into 4 information levels corresponding to (1) building and earthquake characteristics, (2) building response and engineering demand parameters, (3) ordinate and spectral values of intensity measures, and (4) duration of ground motion.
Building and earthquake characteristics
Building information (i.e. geographical coordinates, number of floors, structural material) is summarized in Table 1.1, as given by the BRI and CESMD websites (network_ID).
Building_ID is the unique code of each building based on the nomenclature of data providers. B_lat and B_long correspond to the latitude and longitude of the building location, given in decimal degrees.
Vs30 is the equivalent shear wave velocity for the first 30 meters of the uppermost soil layer. Vs30 was obtained from the Japan Seismic Hazard Information Station (j-shis.bosai.go.jp) and the USGS Vs30 map viewer application (http://usgs.maps.arcgis.com).
Height is the distance between the top and bottom sensors, used to compute structural drift. Floors is the total number of floors in the building, excluding basements and penthouses. Material describes the material of the structural elements of the building: Reinforced Concrete (RC), Steel (S), Steel-Reinforced Concrete (SRC), Masonry (M) and Wood (W), completed by the structural system when available.
The NDE1.0 flat-file provides the pre-seismic fi and co-seismic fmin fundamental frequency values required to compute the spectral IM and to analyze nonlinear elastic processes in the following chapters. Values of fi and fmin were computed as explained in the section 1.3.
Intensity of ground motion
The NDE1.0 flat-file includes several ground motion intensity parameters, classified as ordinary and spectral intensity measures (Tables 1.4 and 1.5, respectively). In addition to PGA, PGV and PGD, Arias Intensity (i.e. Ia, Arias, 1970), Destructive Potential (DP, Araya and Saragoni, 1984), and Cumulative Absolute Velocity (i.e. CAV, EPRI, 1988) are computed from the acceleration time histories a(t). Ia is an energy-based parameter that includes both amplitude and duration of the seismic shaking. It is often linked to the cumulative damage experienced by a structure, where damage is considered to be proportional to the energy dissipated per unit weight during the overall duration of the motion. Ia is defined as:
where g is the acceleration due to gravity and tf is the total duration of the recording. DP is a modification of Ia, where the frequency content of the earthquake is considered, as follows:
where v02 is the number of zero crossings per unit of time. CAV is assumed to reflect the damaging potential of seismic loading. CAV is given by:
In Table 1.5, spectral value-based IMs are provided for a 5% damping ratio. The frequency information suggests that spectral values should be more closely related to damage potential than peak values. We use the algorithms given by Papazafeiropoulos (2015), based on Newmark and Hall (1982) to generate response or pseudo-response spectra values, considering both frequencies: pre-seismic (Sa1, Sv1 and Sd1) and co-seismic (Sa2, Sv2 and Sd2) frequencies for acceleration, velocity and displacement, respectively. In order to take into consideration the frequency shift during the seismic loading, the NDE1.0 flat-file also includes the mean spectral values computed between fmin and fi (Avg_Sa, Avg_Sv and Avg_Sd). This approach has been used by Bommer et al. (2004) and Perrault and Guéguen (2015) to take into account the co-seismic nonlinear response of buildings, also reducing the uncertainties in the prediction of EDP (Perrault and Guéguen, 2015).
Figure 1.9 shows the prediction of co-seismic frequency fmin as a function of PGA, according to functional form (Eq. 1.10), assuming fmin as a proxy of the co-seismic capacity of the structure related to non-ductile/ductile transition (Table A2 in the appendix summarizes all the σ values). The variability in the prediction of fmin for all Japanese data (Fig. 1.9a) is 0.212, which is reduced to 0.086 and 0.048 when the data are sorted by material type, such as SRC (Fig. 1.9c) or S (Fig. 1.9d). For RC buildings (Fig. 1.9b), the data are more scattered, due to different characteristics (i.e. number of floors, different structural systems, such as shear-walls, frames, etc.) but also because heterogeneous and fractured RC materials are more sensitive to loading. Indeed, several authors have demonstrated the influence of the microstructure on nonlinear elastic behavior, proving that granular/heterogeneous materials (e.g., rocks, concrete, damaged materials) show a wide range of nonlinear elastic responses (Guyer and Johnson, 1999; TenCate et al., 2000; Rivière et al., 2015 and 2016). This was also confirmed in by Astorga et al. (2019) for civil engineering structures.
In table A2, the results of the variability of functional form suggest that considering fundamental frequency as EDP yields a lower variability than ∆, suggesting that fundamental frequency could be used for predictive models of structural response. Finally, by testing and combining the most effective IMs, further studies could be conducted on P(EDP | IM) in an attempt to reduce EDP prediction uncertainties, such as structural drift or co-seismic resonance frequency.
Figure 1.9 P(fmin|PGA) models. a) all Japanese data, b) Japanese RC buildings, c) Japanese SRC buildings, and d) Japanese steel buildings. The solid and dashed lines correspond to the mean ± σ of the functional form indicated in the title of the figures. values are also indicated. The Y-axis corresponds to values of fmin normalized by the minimum value of each data group.
Uncertainties related to building-specific damage prediction
For structural performance analysis, Luco and Cornell (2007) defined a sufficient IM as the ground motion parameter that renders building response, then associated seismic damage, conditionally independent of earthquake magnitude and source-to-building distance. With respect to seismic ground motion variability, most GMPEs activities have focused on understanding its origin, making the most of the large amount of data published over the past decade. Al Atik et al. (2010) described the uncertainties of seismic ground motion by decomposing total variability into between-event (τ) and within-event (ϕ) variability. Along the same lines as Al Atik et al., (2010), a preliminary analysis is presented herein to examine the origin of the variability in the structural response, using the so-called Building Damage Prediction Equation (BDPE), first introduced by Perrault and Guéguen (2015) and corresponding to Eq.
Table of contents :
1. NDE1.0 – A NEW DATABASE OF EARTHQUAKE DATA RECORDINGS FROM BUILDINGS FOR ENGINEERING APPLICATIONS
1.1. INTRODUCTION
1.2. DATA DESCRIPTION
1.3. DATA PROCESSING: THE WIGNER-VILLE DISTRIBUTION
1.4. STRUCTURE OF THE NDE1.0 FLAT-FILE
1.4.1. Building and earthquake characteristics
1.4.2. Building response and engineering demand parameters
1.4.3. Intensity of ground motion
1.4.4. Strong motion duration
1.5. EMPIRICAL PREDICTION OF BUILDING RESPONSE P(EDP|IM) AND ASSOCIATED UNCERTAINTIES
1.6. UNCERTAINTIES RELATED TO BUILDING-SPECIFIC DAMAGE PREDICTION
1.7. SEISMIC VULNERABILITY ASSESSMENT AND PERFORMANCE-BASED ANALYSIS
1.8. CONCLUSIONS
2. NONLINEAR ELASTICITY OBSERVED IN BUILDINGS: THE CASE OF THE ANX BUILDING (JAPAN)
2.1. INTRODUCTION
2.2. DESCRIPTION OF THE ANX BUILDING AND DATA
2.3. DATA PROCESSING
2.4. VARIATION OF THE RESONANCE FREQUENCY
2.5. DISCUSSION OF THE ORIGIN OF THE NONLINEARITIES
2.6. OBSERVATIONS IN ANOTHER BUILDING: THE CASE OF THE THU BUILDING
2.7. CONCLUSIONS
3. SLOW DYNAMICS (RECOVERY) USED AS A PROXY FOR SEISMIC STRUCTURAL HEALTH MONITORING
3.1. INTRODUCTION
3.2. METHODOLOGY
3.3. RELAXATION MODELS
3.3.1. Relaxation function
3.3.2. Relaxation time spectrum
3.4. EVOLUTION OF RELAXATION PARAMETERS OVER TIME
3.5. RELAXATION PARAMETERS AND STRUCTURAL DAMAGE
3.6. THE MULTI-SCALE FEATURE OF FREQUENCY RECOVERY IN BUILDINGS
3.6.1. Backbone recovery curve and hysteresis during aftershocks
3.6.2. Relaxation models applied to long-term structural recovery
3.7. CONCLUSIONS
4. NONLINEAR ELASTIC RESPONSE TO MONITOR STRUCTURAL DAMAGE IN BUILDINGS OF DIFFERENT TYPOLOGIES
4.1. INTRODUCTION
4.2. FREQUENCY VARIATIONS OVER TIME
4.3. EVOLUTION OF RELAXATION PARAMETERS
4.4. INFLUENCE OF LOADING AND LOADING RATE
4.4.1. Case 1: the ANX building
4.4.2. Case 2: the THU building
4.5. CONCLUSIONS
GENERAL CONCLUSIONS
